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Motivated by Andy Weir's novel Hail Mary, I was just reading an answer regarding the time needed to travel between stars with a constant acceleration (and deceleration after the mid-way point). Due to time and space dilation, travel times become surprisingly short: For example, traveling to Kepler 186f that is 490 light years away would only take subjective 12.1 years. Since nothing can travel faster than light this implies that due to space dilation, the star must be much closer in the ship's reference frame at maximum speed; by necessity, closer than 12.1/2 light years. (I assume that because the ship is fastest at mid point, and the distance between Earth and star is shortest there, at maximum speed; the rest of the voyage is less efficient but the trip still takes no longer than 12.1 years. If the ship could continue to coast close to light speed from there (and then zoom by the star and not try to stay there), it would arrive much earlier than if it decelerated during the second half of the trip. Therefore, the star must be closer than 6.05 light years at that point, and the distance star-Sol would be < 12.1 light years.)

Now the star is, by observers on the ship, measured to be 490 light years away when the ship starts — but less than 12.1/2 at mid point. I suppose that is measured as the star (and everything else) moving by observers on that ship (and thus emitting blue-shifted light), just like we are seeing distant stars moving away from us in the expanding universe.

Didn't Kepler 186 just travel 472 or so light years in 6.05 years?

Peter - Reinstate Monica
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2 Answers2

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A moment ago, Canada was 1000 miles to my right. I just turned around (in less than a tenth of a second) and now it's 1000 miles to my left. Did Canada move 2000 miles in a tenth of a second?

No, actually Canada is and always has been and always will be 1000 miles to my right in my initial frame, and is and always has been and always will be 1000 miles to my left in my current frame. The fact that I changed frames doesn't tell you a thing you didn't already know about Canada.

Likewise, Kepler 186 (which, unlike Canada, might eventually move substantially relative to the earth) is and has for a very long time been, and will for a very long time be, about 490 light years from earth in the earth's frame. It is some other distance from the earth (and some still other distance from you) in your instantaneous inertial frame at the midpoint of your journey. This is exactly as remarkable as my experience with Canada.

WillO
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You mention in a comment: "I thought that in general relativity an accelerated frame is a valid reference frame as well."

General relativity is more subtle than that.

The introduction of GR was a revolution, a profound rethinking, as much of a revolution as the transition from newtonian theory of motion to special relativity. But the change brought about by the introduction of GR was not in the direction that your comment supposes.


(This is a comment, but I posted it in answer space because I want to separate into paragraph, and by design: in comment space newlines are not available.)

Cleonis
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